Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Equivalent Extensions to Caristi-Kirk’s Fixed Point Theorem, Ekeland’s Variational Principle, and Takahashi’s Minimization Theorem | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 970579 20 pages doi 2010 970579 Research Article Equivalent Extensions to Caristi-Kirk s Fixed Point Theorem Ekeland s Variational Principle and Takahashi s Minimization Theorem Zili Wu Department of Mathematical Sciences Xi an Jiaotong-Liverpool University 111 Ren Ai Road Dushu Lake Higher Education Town Suzhou Industrial Park Suzhou Jiangsu 215123 China Correspondence should be addressed to Zili Wu ziliwu@ Received 26 September 2009 Accepted 24 November 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 Zili Wu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. With a recent result of Suzuki 2001 we extend Caristi-Kirk s fixed point theorem Ekeland s variational principle and Takahashi s minimization theorem in a complete metric space by replacing the distance with a T-distance. In addition these extensions are shown to be equivalent. When the T-distance is . in its second variable they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds which are in turn useful for extending some existing results such as the petal theorem. 1. Introduction Let X d be a complete metric space and f X -TO to a proper lower semicontinuous . bounded below function. Caristi-Kirk fixed point theorem 1 Theorem states that there exists x0 6 Tx0 for a relation or multivalued mapping T X X if for each x e X with infXf f x there exists X e Tx such that d x x f X f x see also 2 Theorem or 3 Theorem C while Ekeland s variational principle EVP 4 5 asserts that for each e e 0 TO and u e X with f u infXf e there exists v e X such that f v f u and f x ed v x f v Vx e X with x fv. EVP has been shown to have many equivalent formulations such as .